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 KelvinBer

 http://functions.wolfram.com/03.18.21.0002.01

 Input Form

 Integrate[(t^(\[Alpha] - 1) KelvinBer[\[Nu], t])/E^(p t), {t, 0, Infinity}] == (1/Gamma[1 + \[Nu]]) (2^(-2 - \[Nu]) p^(-\[Alpha] - \[Nu]) Gamma[\[Alpha] + \[Nu]] (4 Cos[(3 Pi \[Nu])/4] HypergeometricPFQ[{\[Alpha]/4 + \[Nu]/4, 1/4 + \[Alpha]/4 + \[Nu]/4, 1/2 + \[Alpha]/4 + \[Nu]/4, 3/4 + \[Alpha]/4 + \[Nu]/4}, {1/2, 1/2 + \[Nu]/2, 1 + \[Nu]/2}, -(1/p^4)] - (1/(p^2 (1 + \[Nu]))) ((\[Alpha] + \[Nu]) (1 + \[Alpha] + \[Nu]) HypergeometricPFQ[{1/2 + \[Alpha]/4 + \[Nu]/4, 3/4 + \[Alpha]/4 + \[Nu]/4, 1 + \[Alpha]/4 + \[Nu]/4, 5/4 + \[Alpha]/4 + \[Nu]/4}, {3/2, 1 + \[Nu]/2, 3/2 + \[Nu]/2}, -(1/p^4)] Sin[(3 Pi \[Nu])/4]))) /; Re[\[Alpha] + \[Nu]] > 0 && Re[p] > 1/Sqrt[2]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p"]], " ", "t"]]], RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["p", "4"]]]]]], "]"]]]], "-", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["p", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Nu]"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["p", "4"]]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]]]], ")"]]]]]], ")"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Nu]"]], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "p", "]"]], ">", FractionBox["1", SqrtBox["2"]]]]]]]]]]

 MathML Form

 0 t α - 1 - p t ber ν ( t ) t 2 - ν - 2 p - α - ν Γ ( α + ν ) Γ ( ν + 1 ) ( 4 cos ( 3 π ν 4 ) 4 F 3 ( α 4 + ν 4 , α 4 + ν 4 + 1 4 , α 4 + ν 4 + 1 2 , α 4 + ν 4 + 3 4 ; 1 2 , ν 2 + 1 2 , ν 2 + 1 ; - 1 p 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox["F", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"], "+", FractionBox["1", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", SuperscriptBox["p", "4"]]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] - ( α + ν ) ( α + ν + 1 ) sin ( 3 π ν 4 ) p 2 ( ν + 1 ) 4 F 3 ( α 4 + ν 4 + 1 2 , α 4 + ν 4 + 3 4 , α 4 + ν 4 + 1 , α 4 + ν 4 + 5 4 ; 3 2 , ν 2 + 1 , ν 2 + 3 2 ; - 1 p 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox["F", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"], "+", FractionBox["5", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", SuperscriptBox["p", "4"]]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] ) /; Re ( α + ν ) > 0 Re ( p ) > 1 2 Condition t 0 t α -1 -1 p t KelvinBer ν t 2 -1 ν -2 p -1 α -1 ν Gamma α ν Gamma ν 1 -1 4 3 ν 4 -1 HypergeometricPFQ α 4 -1 ν 4 -1 α 4 -1 ν 4 -1 1 4 α 4 -1 ν 4 -1 1 2 α 4 -1 ν 4 -1 3 4 1 2 ν 2 -1 1 2 ν 2 -1 1 -1 1 p 4 -1 -1 α ν α ν 1 3 ν 4 -1 p 2 ν 1 -1 HypergeometricPFQ α 4 -1 ν 4 -1 1 2 α 4 -1 ν 4 -1 3 4 α 4 -1 ν 4 -1 1 α 4 -1 ν 4 -1 5 4 3 2 ν 2 -1 1 ν 2 -1 3 2 -1 1 p 4 -1 α ν 0 p 1 2 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p_"]], " ", "t_"]]], " ", RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["p", "4"]]]]]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Nu]"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Alpha]", "4"], "+", FractionBox["\[Nu]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["p", "4"]]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]]]], RowBox[List[SuperscriptBox["p", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Nu]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "p", "]"]], ">", FractionBox["1", SqrtBox["2"]]]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02